Let $X$ be a [Banach space](/page/Banach%20Space) over $\mathbb R$ or $\mathbb C$ with a normalized Schauder basis $(e_k)_{k\ge 1}$, let $\mathcal D=\{e_k:k\ge 1\}$, and write $f=\sum_{k=1}^{\infty} a_k e_k$. For a finite set $A\subset \mathbb N$, write $P_A f=\sum_{k\in A}a_k e_k$. A set $A$ is a thresholding set of size $m$ for $f$ if $|A|=m$ and

\begin{align*} \min_{k\in A}|a_k|\ge \sup_{j\notin A}|a_j|. \end{align*}

Suppose the basis is quasi-greedy with constant $K$,

\begin{align*} \|P_A f\|_X\le K\|f\|_X \end{align*}

whenever $A$ is a thresholding set for $f$, and democratic with constant $\Delta$,

\begin{align*} \left\|\sum_{k\in A}\varepsilon_k e_k\right\|_X\le \Delta \left\|\sum_{k\in B}\eta_k e_k\right\|_X \end{align*}

whenever $|A|\le |B|$ and $|\varepsilon_k|=|\eta_k|=1$. Let $G_m(f)=P_{\Lambda_m(f)}f$, where $\Lambda_m(f)$ is a chosen set of $m$ indices whose coefficients have largest magnitudes, with an arbitrary deterministic tie-breaking rule. Then there exists a constant $C\ge 1$, depending only on $K$ and $\Delta$, such that for all $f\in X$ and all $n\ge 1$,

\begin{align*} \|f-G_n(f)\|_X \le C\, \sigma_n(f)_{X,\mathcal D}. \end{align*}

Equivalently, a normalized Schauder basis is almost greedy precisely when it is quasi-greedy and democratic.